# Uncorrelatedness for random elements of finitely generated groups?

Suppose $$G$$ is a finitely generated group, $$A$$ is its finite set of generators. Lets denote the metric induced by the Cayley graph $$Cay(G, A)$$ on $$G$$ as $$d$$.

Suppose $$\{X_i\}_{n = 0}^\infty$$ is a sequence of i.i.d. random elements in $$G$$ and $$\{Y_i\}_{n = 0}^\infty$$ is another sequence of i.i.d. random elements in $$G$$ (the distributions of $$X_0$$ and $$Y_0$$ may still be different), that satisfy the following properties:

1. $$\forall n, m \in \mathbb{N}$$ $$X_n$$ and $$Y_m$$ are mutually independant.

2. $$d(X_0, e)$$ and $$d(Y_0, e)$$ both have finite second moments

Does it follow from that, that exists $$r \in \mathbb{R}$$ (the same for all $$G$$, $$A$$, $$\{X_i\}_{n = 0}^\infty$$ and $$\{Y_i\}_{n = 0}^\infty$$), such that $$P(\inf_{g, h \in G}\overline{\lim_{n \to \infty}} \frac{d(\Pi_{j = 0}^nX_j, g)d(\Pi_{j = 0}^nY_j, h)}{n} \leq r) = 1$$?

I managed to prove this fact only for finite groups (which is rather trivial):

$$\exists C \in \mathbb{N} \forall g, h \in G, d(g, h) < C$$ $$P(\inf_{g, h \in G}\overline{\lim_{n \to \infty}} \frac{d(X_n, g)d(Y_n, h)}{n} \leq \overline{\lim_{n \to \infty}} \frac{C^2}{n} \leq 0) = 1$$

This question was inspired by the theorem that says that random variables with finite second moment are uncorrelated.

First, I thought, that $$r = 0$$, but the counterexamples to that statement seem to be easily constructed for $$G = \langle a \rangle_\infty$$, $$A = \{a\}$$.

• What does i.i.d. stand for? – user1729 Apr 18 at 16:12
• @user1729, "i.i.d" stands for "independent and identically distributed" – Yanior Weg Apr 18 at 16:14